9 MEASURABLE FUNCTIONS 137 (13.9.8) A function equivalent to a measurable function is measurable. This follows from (13.9.4) and the fact that if K is compact and N c K is negligible, then there exists a compact set K' c K — N with measure arbitrarily close to the measure of K (13.7.9). A mapping f defined almost everywhere in X, with values in a topological space Y, is said to be measurable if every mapping of X into Y which is equivalent to/ (13.6) is measurable. Clearly (13.9.8) it is enough for one such mapping to be measurable. (13.9.8.1) In particular, it follows, from (13.9.6) and (13.9.8) that if /and g (with values in K) are defined and finite almost everywhere inX and measurable, then/+ g and/# (which are defined almost everywhere) are measurable. (13.9.9) Let u be a measurable mapping 0/X into a topological space Y.IfM is any open or closed subset ofY, then u^i(M) is measurable. Consider a partition of X consisting of a sequence of compact sets Kn and a negligible set N, such that the restrictions u \ Kn are continuous. If M c Y is closed, then u~1(M) n Kn is compact (3.11.4), hence tT^M) is measurable. If M is open, then X — w^M) = u~1(Y — M) is measurable, and therefore so is u~*(M) (13.9.3). (13.9.10) (Egoroff's theorem) Let Y be a metric space and let (/„) be a sequence of measurable mappings ofX into Y such that, for almost all x e X, the sequence (/,(*)) converges to a limit f(x). Then ( 1 ) fis measurable ; (2) for each compact subset KofX and each s > 0, there exists a compact subset K' of K such that ^(K - K') :g £ and such that the restrictions fn \ K' are continuous and converge uniformly in K' tof\ K'. Clearly it is sufficient to prove the second assertion ((13.9.4) and (7.2.1)). Let K0 be a compact subset of K such that ju(K — K0) ^ le and such that the restrictions/, | K0 are continuous for all n (13.9.5). If d denotes the distance function on Y, let BM>r be the set of all x e K0 such that </(/,(*), /«(*)) £ 1/r for at least one pair of integers (p, q) such that p 2> n and q ^ n. If Aptqtt, is the set of points x e K0 such that d(fp(x)9fq(xj) ^ 1/r, then we havee restrictions u \ Ln is continuous.